25.1.10 Pluripolar sets and the plurifine topology1AFE6A788B80

Josefson proved that locally pluripolar subsets of $\mathbb C^ n$ are globally pluripolar [Jos78]; see theorem 5.47.10. The approach through capacity is due to Bedford and Taylor [BT82]; for relative extremal functions compare [Theorem 4.32, pp. 118–119, Theorem 4.34, pp. 119–121, and Theorem 4.40 and Corollary 4.41, pp. 124–125, GZ17]. The identities for the masses of relative extremal functions of compact and of open sets (proposition 5.47.6, proposition 5.47.7) are proved with the homogeneous balayage of corollary 7.7.5; together with approximation by decreasing open sets they give the characterization of pluripolar sets by capacity, theorem 5.47.8.

For the local base of plurifine neighborhoods in proposition 5.47.15 compare [arXiv version, Lemma 3.1, pp. 3–4, EMW06]. The main theorem of that paper, the local connectedness of the plurifine topology, is not treated here; its proof also uses finely subharmonic functions. The plurifine quasi-Lindelöf principle, theorem 5.47.20, follows from Choquet's lemma and the pluripolarity of the exceptional sets of upper envelopes (corollary 5.47.11). The plurifine locality of Bedford–Taylor products, theorem 5.47.22, is deduced from their behavior under maxima (proposition 7.3.10), up to exceptional sets that products of bounded functions do not charge. In subsection 5.47.24 the difference of two plurisubharmonic functions, and its quasi-extrema, are considered only outside their common pole set, where the difference is defined.